This entry is about the formal dual to tensoring in the generality of category theory. For the different concept of cotensor product of comodules see there.
In a closed symmetric monoidal category $V$ the internal hom $[-,-] : V^{op} \times V \to V$ satisfies the natural isomorphism
for all objects $v_i \in V$. If we regard $V$ as a $V$-enriched category we write $V(v_1,v_2) := [v_1,v_2]$ and this reads
If we now pass more generally to any $V$-enriched category $C$ then we still have the enriched hom object functor $C(-,-) : C^{op} \times C \to V$. One says that $C$ is powered over $V$ if it is in addition equipped also with a mixed operation $\pitchfork : V^{op} \times C \to C$ such that $\pitchfork(v,c)$ behaves as if it were a hom of the object $v \in V$ into the object $c \in C$ in that it satisfies the natural isomorphism
Let $V$ be a closed symmetric monoidal category. In a $V$-enriched category $C$, the power of an object $y\in C$ by an object $v\in V$ is an object $\pitchfork(v,y) \in C$ with a natural isomorphism
where $C(-,-)$ is the $V$-valued hom of $C$ and $V(-,-)$ is the internal hom of $V$.
We say that $C$ is powered or cotensored over $V$ if all such power objects exist.
Powers are frequently called cotensors and a $V$-category having all powers is called cotensored, while the word “power” is reserved for the case $V=$ Set. However, there seems to be no good reason for making this distinction. Moreover, the word “tensor” is fairly overused, and unfortunate since a tensor (= a copower) is a colimit, while a cotensor (= power) is a limit.
$V$ itself is always powered over itself, with $\pitchfork(v_1,v_2) := [v_1,v_2]$.
Every locally small category $C$ ($V = (Set,\times)$ ) with all products is powered over Set: the powering operation
of an object $c$ by a set $S$ forms the $|S|$-fold cartesian product of $c$ with itself, where $|S|$ is the cardinality of $S$.
The defining natural isomorphism
is effectively the definition of the product (see limit).
power
Section 3.7 of
Section 6.5 of